1 /* mpz_probab_prime_p -- 2 An implementation of the probabilistic primality test found in Knuth's 3 Seminumerical Algorithms book. If the function mpz_probab_prime_p() 4 returns 0 then n is not prime. If it returns 1, then n is 'probably' 5 prime. If it returns 2, n is surely prime. The probability of a false 6 positive is (1/4)**reps, where reps is the number of internal passes of the 7 probabilistic algorithm. Knuth indicates that 25 passes are reasonable. 8 9 Copyright 1991, 1993, 1994, 1996-2002, 2005 Free Software Foundation, Inc. 10 11 This file is part of the GNU MP Library. 12 13 The GNU MP Library is free software; you can redistribute it and/or modify 14 it under the terms of either: 15 16 * the GNU Lesser General Public License as published by the Free 17 Software Foundation; either version 3 of the License, or (at your 18 option) any later version. 19 20 or 21 22 * the GNU General Public License as published by the Free Software 23 Foundation; either version 2 of the License, or (at your option) any 24 later version. 25 26 or both in parallel, as here. 27 28 The GNU MP Library is distributed in the hope that it will be useful, but 29 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 30 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 31 for more details. 32 33 You should have received copies of the GNU General Public License and the 34 GNU Lesser General Public License along with the GNU MP Library. If not, 35 see https://www.gnu.org/licenses/. */ 36 37 #include "gmp.h" 38 #include "gmp-impl.h" 39 #include "longlong.h" 40 41 static int isprime (unsigned long int); 42 43 44 /* MPN_MOD_OR_MODEXACT_1_ODD can be used instead of mpn_mod_1 for the trial 45 division. It gives a result which is not the actual remainder r but a 46 value congruent to r*2^n mod d. Since all the primes being tested are 47 odd, r*2^n mod p will be 0 if and only if r mod p is 0. */ 48 49 int 50 mpz_probab_prime_p (mpz_srcptr n, int reps) 51 { 52 mp_limb_t r; 53 mpz_t n2; 54 55 /* Handle small and negative n. */ 56 if (mpz_cmp_ui (n, 1000000L) <= 0) 57 { 58 int is_prime; 59 if (mpz_cmpabs_ui (n, 1000000L) <= 0) 60 { 61 is_prime = isprime (mpz_get_ui (n)); 62 return is_prime ? 2 : 0; 63 } 64 /* Negative number. Negate and fall out. */ 65 PTR(n2) = PTR(n); 66 SIZ(n2) = -SIZ(n); 67 n = n2; 68 } 69 70 /* If n is now even, it is not a prime. */ 71 if ((mpz_get_ui (n) & 1) == 0) 72 return 0; 73 74 #if defined (PP) 75 /* Check if n has small factors. */ 76 #if defined (PP_INVERTED) 77 r = MPN_MOD_OR_PREINV_MOD_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) PP, 78 (mp_limb_t) PP_INVERTED); 79 #else 80 r = mpn_mod_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) PP); 81 #endif 82 if (r % 3 == 0 83 #if GMP_LIMB_BITS >= 4 84 || r % 5 == 0 85 #endif 86 #if GMP_LIMB_BITS >= 8 87 || r % 7 == 0 88 #endif 89 #if GMP_LIMB_BITS >= 16 90 || r % 11 == 0 || r % 13 == 0 91 #endif 92 #if GMP_LIMB_BITS >= 32 93 || r % 17 == 0 || r % 19 == 0 || r % 23 == 0 || r % 29 == 0 94 #endif 95 #if GMP_LIMB_BITS >= 64 96 || r % 31 == 0 || r % 37 == 0 || r % 41 == 0 || r % 43 == 0 97 || r % 47 == 0 || r % 53 == 0 98 #endif 99 ) 100 { 101 return 0; 102 } 103 #endif /* PP */ 104 105 /* Do more dividing. We collect small primes, using umul_ppmm, until we 106 overflow a single limb. We divide our number by the small primes product, 107 and look for factors in the remainder. */ 108 { 109 unsigned long int ln2; 110 unsigned long int q; 111 mp_limb_t p1, p0, p; 112 unsigned int primes[15]; 113 int nprimes; 114 115 nprimes = 0; 116 p = 1; 117 ln2 = mpz_sizeinbase (n, 2); /* FIXME: tune this limit */ 118 for (q = PP_FIRST_OMITTED; q < ln2; q += 2) 119 { 120 if (isprime (q)) 121 { 122 umul_ppmm (p1, p0, p, q); 123 if (p1 != 0) 124 { 125 r = MPN_MOD_OR_MODEXACT_1_ODD (PTR(n), (mp_size_t) SIZ(n), p); 126 while (--nprimes >= 0) 127 if (r % primes[nprimes] == 0) 128 { 129 ASSERT_ALWAYS (mpn_mod_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) primes[nprimes]) == 0); 130 return 0; 131 } 132 p = q; 133 nprimes = 0; 134 } 135 else 136 { 137 p = p0; 138 } 139 primes[nprimes++] = q; 140 } 141 } 142 } 143 144 /* Perform a number of Miller-Rabin tests. */ 145 return mpz_millerrabin (n, reps); 146 } 147 148 static int 149 isprime (unsigned long int t) 150 { 151 unsigned long int q, r, d; 152 153 if (t < 3 || (t & 1) == 0) 154 return t == 2; 155 156 for (d = 3, r = 1; r != 0; d += 2) 157 { 158 q = t / d; 159 r = t - q * d; 160 if (q < d) 161 return 1; 162 } 163 return 0; 164 } 165